let V, A be set ; :: thesis: for a, b being Element of V
for x0, y0 being Nat st not V is empty & A is_without_nonatomicND_wrt V & a <> b & A is complex-containing & ( for d being TypeSCNominativeData of V,A holds a is_complex_on d ) & ( for d being TypeSCNominativeData of V,A holds b is_complex_on d ) holds
<*(gcd_inv (V,A,a,b,x0,y0)),(gcd_main_loop (V,A,a,b)),(PP_and ((Equality (A,a,b)),(gcd_inv (V,A,a,b,x0,y0))))*> is SFHT of (ND (V,A))
let a, b be Element of V; :: thesis: for x0, y0 being Nat st not V is empty & A is_without_nonatomicND_wrt V & a <> b & A is complex-containing & ( for d being TypeSCNominativeData of V,A holds a is_complex_on d ) & ( for d being TypeSCNominativeData of V,A holds b is_complex_on d ) holds
<*(gcd_inv (V,A,a,b,x0,y0)),(gcd_main_loop (V,A,a,b)),(PP_and ((Equality (A,a,b)),(gcd_inv (V,A,a,b,x0,y0))))*> is SFHT of (ND (V,A))
let x0, y0 be Nat; :: thesis: ( not V is empty & A is_without_nonatomicND_wrt V & a <> b & A is complex-containing & ( for d being TypeSCNominativeData of V,A holds a is_complex_on d ) & ( for d being TypeSCNominativeData of V,A holds b is_complex_on d ) implies <*(gcd_inv (V,A,a,b,x0,y0)),(gcd_main_loop (V,A,a,b)),(PP_and ((Equality (A,a,b)),(gcd_inv (V,A,a,b,x0,y0))))*> is SFHT of (ND (V,A)) )
set p = gcd_inv (V,A,a,b,x0,y0);
set r = PP_not (Equality (A,a,b));
set f = gcd_loop_body (V,A,a,b);
assume ( not V is empty & A is_without_nonatomicND_wrt V & a <> b & A is complex-containing & ( for d being TypeSCNominativeData of V,A holds a is_complex_on d ) & ( for d being TypeSCNominativeData of V,A holds b is_complex_on d ) ) ; :: thesis: <*(gcd_inv (V,A,a,b,x0,y0)),(gcd_main_loop (V,A,a,b)),(PP_and ((Equality (A,a,b)),(gcd_inv (V,A,a,b,x0,y0))))*> is SFHT of (ND (V,A))
then <*(gcd_inv (V,A,a,b,x0,y0)),(gcd_loop_body (V,A,a,b)),(gcd_inv (V,A,a,b,x0,y0))*> is SFHT of (ND (V,A)) by Th21;
then A2: <*(PP_and ((PP_not (Equality (A,a,b))),(gcd_inv (V,A,a,b,x0,y0)))),(gcd_loop_body (V,A,a,b)),(gcd_inv (V,A,a,b,x0,y0))*> is SFHT of (ND (V,A)) by NOMIN_3:4, NOMIN_3:15;
<*(PP_inversion (gcd_inv (V,A,a,b,x0,y0))),(gcd_loop_body (V,A,a,b)),(gcd_inv (V,A,a,b,x0,y0))*> is SFHT of (ND (V,A)) by NOMIN_3:19;
then <*(PP_and ((PP_not (Equality (A,a,b))),(PP_inversion (gcd_inv (V,A,a,b,x0,y0))))),(gcd_loop_body (V,A,a,b)),(gcd_inv (V,A,a,b,x0,y0))*> is SFHT of (ND (V,A)) by NOMIN_3:4, NOMIN_3:15;
hence <*(gcd_inv (V,A,a,b,x0,y0)),(gcd_main_loop (V,A,a,b)),(PP_and ((Equality (A,a,b)),(gcd_inv (V,A,a,b,x0,y0))))*> is SFHT of (ND (V,A)) by A2, NOMIN_3:26; :: thesis: verum